@article{BraunFeudelSeehafer1997, author = {Braun, Robert and Feudel, Fred and Seehafer, Norbert}, title = {Bifurcations and chaos in an array of forced vortices}, year = {1997}, language = {en} } @article{BrownCanfieldFieldetal.1999, author = {Brown, M. R. and Canfield, R. C. and Field, G. and Kulsrud, R. and Pevtsov, A. A. and Rosner, R. and Seehafer, Norbert}, title = {Magnetic helicity in space and laboratory plasmas: Editorial summary}, year = {1999}, language = {en} } @article{DemircanScheelSeehafer2000, author = {Demircan, Ayhan and Scheel, S. and Seehafer, Norbert}, title = {Heteroclinic behavior in rotating Rayleigh-Benard convection}, year = {2000}, abstract = {We investigate numerically the appearance of heteroclinic behavior in a three-dimensional, buoyancy-driven fluid layer with stress-free top and bottom boundaries, a square horizontal periodicity with a small aspect ratio, and rotation at low to moderate rates about a vertical axis. The Prandtl number is 6.8. If the rotation is not too slow, the skewed-varicose instability leads from stationary rolls to a stationary mixed-mode solution, which in turn loses stability to a heteroclinic cycle formed by unstable roll states and connections between them. The unstable eigenvectors of these roll states are also of the skewed-varicose or mixed-mode type and in some parameter regions skewed-varicose like shearing oscillations as well as square patterns are involved in the cycle. Always present weak noise leads to irregular horizontal translations of the convection pattern and makes the dynamics chaotic, which is verified by calculating Lyapunov exponents. In the nonrotating case the primary rolls lose, depending on the aspect ratio, stability to traveling waves or a stationary square pattern. We also study the symmetries of the solutions at the intermittent fixed points in the heteroclinic cycle.}, language = {en} } @article{DemircanSeehafer2001, author = {Demircan, Ayhan and Seehafer, Norbert}, title = {Dynamos in rotating and nonrotating convection in the form of asymmetric squares}, year = {2001}, abstract = {We study the dynamo properties of asymmetric square patterns in Boussinesq Rayleigh-B'enard convection in a plane horizontal layer. Cases without rotation and with weak rotation about a vertical axis are considered. There exist different types of solutions distinguished by their symmetry, among them such with flows possessing a net helicity and being capable of kinematic dynamo action in the presence as well as in the absence of rotation. In the nonrotating case these flows are, however, always only kinematic, not nonlinear dynamos. Nonlinearly the back-reaction of the magnetic field then forces the solution into the basin of attraction of a roll pattern incapable of dynamo action. But with rotation added parameter regions are found where the Coriolis force counteracts the Lorentz force in such a way that the asymmetric squares are also nonlinear dynamos.}, language = {en} } @article{DemircanSeehafer2001, author = {Demircan, Ayhan and Seehafer, Norbert}, title = {Nonlinear square patterns in Rayleigh-Benard convection}, year = {2001}, abstract = {We numerically investigate nonlinear asymmetric square patterns in a horizontal convection layer with up-down reflection symmetry. As a novel feature we find the patterns to appear via the skewed varicose instability of rolls. The time-independent nonlinear state is generated by two unstable checkerboard (symmetric square) patterns and their nonlinear interaction. As the bouyancy forces increase the interacting modes give rise to bifurcations leading to a periodic alternation between a nonequilateral hexagonal pattern and the square pattern or to different kinds of standing oscillations.}, language = {en} } @article{DemircanSeehafer2002, author = {Demircan, Ayhan and Seehafer, Norbert}, title = {Dynamo in asymmetric square convection}, issn = {0309-1929}, year = {2002}, language = {en} } @article{DemircanSeehafer2000, author = {Demircan, Ayhan and Seehafer, Norbert}, title = {Heteroclinic behavior in rotating Rayleigh-B{\´e}nard convection}, year = {2000}, abstract = {We investigate numerically the appearance of heteroclinic behavior in a three-dimensional, buoyancy-driven, rotating fluid layer. Periodic boundary conditions in the horizontal directions and stress-free boundary conditions at the top and bottom are assumed.}, language = {en} } @article{DemircanSeehafer1999, author = {Demircan, Ayhan and Seehafer, Norbert}, title = {Bifurcation to oscillations and chaos in rotating convection}, year = {1999}, language = {en} } @article{DonnerFeudelSeehaferetal.2007, author = {Donner, Reik Volker and Feudel, Fred and Seehafer, Norbert and Sanjuan, Miguel Angel Fernandez}, title = {Hierarchical modeling of a forced Roberts Dynamo}, issn = {0218-1274}, doi = {10.1142/S021812740701941X}, year = {2007}, abstract = {We investigate the dynamo effect in a flow configuration introduced by G. O. Roberts in 1972. Based on a clear energetic hierarchy of Fourier components on the steady-state dynamo branch, an approximate model of interacting modes is constructed covering all essential features of the complete system but allowing simulations with a minimum amount of computation time. We use this model to study the excitation mechanism of the dynamo, the transition from stationary to time-dependent dynamo solutions and the characteristic properties of the latter ones.}, language = {en} } @article{DonnerSeehaferSanjuanetal.2006, author = {Donner, Reik Volker and Seehafer, Norbert and Sanjuan, Miguel Angel Fernandez and Feudel, Fred}, title = {Low-dimensional dynamo modelling and symmetry-breaking bifurcations}, series = {Physica. D, Nonlinear phenomena}, volume = {223}, journal = {Physica. D, Nonlinear phenomena}, number = {2}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0167-2789}, doi = {10.1016/j.physd.2006.08.022}, pages = {151 -- 162}, year = {2006}, abstract = {Motivated by the successful Karlsruhe dynamo experiment, a relatively low-dimensional dynamo model is proposed. It is based on a strong truncation of the magnetohydrodynamic (MHD) equations with an external forcing of the Roberts type and the requirement that the model system satisfies the symmetries of the full MHD system, so that the first symmetry-breaking bifurcations can be captured. The backbone of the Roberts dynamo is formed by the Roberts flow, a helical mean magnetic field and another part of the magnetic field coupled to these two by triadic mode interactions. A minimum truncation model (MTM) containing only these energetically dominating primary mode triads is fully equivalent to the widely used first-order smoothing approximation. However, it is shown that this approach works only in the limit of small wave numbers of the excited magnetic field or small magnetic Reynolds numbers (\$Rm ll 1\$). To obtain dynamo action under more general conditions, secondary mode}, language = {en} } @article{FeudelGellertRuedigeretal.2003, author = {Feudel, Fred and Gellert, Marcus and R{\"u}diger, Sten and Witt, Annette and Seehafer, Norbert}, title = {Dynamo effect in a driven helical flow}, year = {2003}, language = {en} } @article{FeudelRuedigerSeehafer2001, author = {Feudel, Fred and R{\"u}diger, Sten and Seehafer, Norbert}, title = {Bifurcation phenomena and dynamo effect in electrically conducting fluids}, year = {2001}, abstract = {Electrically conducting fluids in motion can act as self-excited dynamos. The magnetic fields of celestial bodies like the Earth and the Sun are generated by such dynamos. Their theory aims at modeling and understanding both the kinematic and dynamic aspects of the underlying processes. Kinematic dynamo models, in which for a prescribed flow the linear induction equation is solved and growth rates of the magnetic field are calculated, have been studied for many decades. But in order to get consistent models and to take into account the back-reaction of the magnetic field on the fluid motion, the full nonlinear system of the magnetohydrodynamic (MHD) equations has to be studied. It is generally accepted that these equations, i.e. the Navier-Stokes equation (NSE) and the induction equation, provide a theoretical basis for the explanation of the dynamo effect. The general idea is that mechanical energy pumped into the fluid by heating or other mechanisms is transferred to the magnetic field by nonlinear interactions. For two special helical flows which are known to be effective kinematic dynamos and which can be produced by appropriate external mechanical forcing, we review the nonlinear dynamo properties found in the framework of the full MHD equations. Specifically, we deal with the ABC flow (named after Arnold, Beltrami and Childress) and the Roberts flow (after G.~O. Roberts). The appearance of generic dynamo effects is demonstrated. Applying special numerical bifurcation-analysis techniques to high-dimensional approximations in Fourier space and varying the Reynolds number (or the strength of the forcing) as the relevant control parameter, qualitative changes in the dynamics are investigated. We follow the bifurcation sequences until chaotic states are reached. The transitions from the primary flows with vanishing magnetic field to dynamo-active states are described in particular detail. In these processes the stagnation points of the flows and their heteroclinic connections play a promoting role for the magnetic field generation. By the example of the Roberts flow we demonstrate how the break up of the heteroclinic lines after the primary bifurcation leads to a complicated intersection of stable and unstable manifolds forming a chaotic web which is in turn correlated with the spatial appearance of the dynamo.}, language = {en} } @article{FeudelSeehafer1995, author = {Feudel, Fred and Seehafer, Norbert}, title = {On the bifurcation phenomena in truncations of the 2D Navier-Stokes equations}, year = {1995}, language = {en} } @article{FeudelSeehafer1995, author = {Feudel, Fred and Seehafer, Norbert}, title = {Bifurcations and pattern formation in two-dimensional Navier-Stokes fluid}, year = {1995}, language = {en} } @article{FeudelSeehaferSchmidtmann1995, author = {Feudel, Fred and Seehafer, Norbert and Schmidtmann, Olaf}, title = {Fluid helicity and dynamo bifurcations}, year = {1995}, language = {en} } @article{FeudelSeehaferSchmidtmann1996, author = {Feudel, Fred and Seehafer, Norbert and Schmidtmann, Olaf}, title = {Bifurcation phenomena of the magnetofluid equations}, year = {1996}, abstract = {We report on bifurcation studies for the incompressible magnetohydrodynamic equations in three space dimensions with periodic boundary conditions and a temporally constant external forcing. Fourier representations of velocity, pressure and magnetic field have been used to transform the original partial differential equations into systems of ordinary differential equations (ODE), to which then special numerical methods for the qualitative analysis of systems of ODE have been applied, supplemented by the simulative calculation of solutions for selected initial conditions. In a part of the calculations, in order to reduce the number of modes to be retained, the concept of approximate inertial manifolds has been applied. For varying (increasing from zero) strength of the imposed forcing, or varying Reynolds number, respectively, time-asymptotic states, notably stable stationary solutions, have been traced. A primary non- magnetic steady state loses, in a Hopf bifurcation, stability to a periodic state with a non-vanishing magnetic field, showing the appearance of a generic dynamo effect. From now on the magnetic field is present for all values of the forcing. The Hopf bifurcation is followed by further, symmetry-breaking, bifurcations, leading finally to chaos. We pay particular attention to kinetic and magnetic helicities. The dynamo effect is observed only if the forcing is chosen such that a mean kinetic helicity is generated; otherwise the magnetic field diffuses away, and the time-asymptotic states are non-magnetic, in accordance with traditional kinematic dynamo theory.}, language = {en} } @article{FeudelSeehaferTuckerman2013, author = {Feudel, Fred and Seehafer, Norbert and Tuckerman, Laurette S.}, title = {Multistability in rotating spherical shell convection}, issn = {1539-3755}, year = {2013}, language = {en} } @article{FeudelSeehaferTuckermanetal.2013, author = {Feudel, Fred and Seehafer, Norbert and Tuckerman, Laurette S. and Gellert, Marcus}, title = {Multistability in rotating spherical shell convection}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {87}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {2}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.87.023021}, pages = {8}, year = {2013}, abstract = {The multiplicity of stable convection patterns in a rotating spherical fluid shell heated from the inner boundary and driven by a central gravity field is presented. These solution branches that arise as rotating waves (RWs) are traced for varying Rayleigh number while their symmetry, stability, and bifurcations are studied. At increased Rayleigh numbers all the RWs undergo transitions to modulated rotating waves (MRWs) which are classified by their spatiotemporal symmetry. The generation of a third frequency for some of the MRWs is accompanied by a further loss of symmetry. Eventually a variety of MRWs, three-frequency solutions, and chaotic saddles and attractors control the dynamics for higher Rayleigh numbers.}, language = {en} } @article{FeudelTuckermanGellertetal.2015, author = {Feudel, Fred and Tuckerman, L. S. and Gellert, M. and Seehafer, Norbert}, title = {Bifurcations of rotating waves in rotating spherical shell convection}, series = {Physical review : E, Statistical, nonlinear and soft matter physics}, volume = {92}, journal = {Physical review : E, Statistical, nonlinear and soft matter physics}, number = {5}, publisher = {American Physical Society}, address = {College Park}, issn = {1539-3755}, doi = {10.1103/PhysRevE.92.053015}, pages = {7}, year = {2015}, abstract = {The dynamics and bifurcations of convective waves in rotating and buoyancy-driven spherical Rayleigh-Benard convection are investigated numerically. The solution branches that arise as rotating waves (RWs) are traced by means of path-following methods, by varying the Rayleigh number as a control parameter for different rotation rates. The dependence of the azimuthal drift frequency of the RWs on the Ekman and Rayleigh numbers is determined and discussed. The influence of the rotation rate on the generation and stability of secondary branches is demonstrated. Multistability is typical in the parameter range considered.}, language = {en} } @article{FeudelTuckermanGellertetal.2015, author = {Feudel, Fred and Tuckerman, L. S. and Gellert, Marcus and Seehafer, Norbert}, title = {Bifurcations of rotating waves in rotating spherical shell convection}, series = {Physical Review E}, volume = {92}, journal = {Physical Review E}, number = {5}, publisher = {American Physical Society}, address = {Woodbury}, issn = {1539-3755}, doi = {10.1103/PhysRevE.92.053015}, year = {2015}, abstract = {The dynamics and bifurcations of convective waves in rotating and buoyancy-driven spherical Rayleigh-Benard convection are investigated numerically. The solution branches that arise as rotating waves (RWs) are traced by means of path-following methods, by varying the Rayleigh number as a control parameter for different rotation rates. The dependence of the azimuthal drift frequency of the RWs on the Ekman and Rayleigh numbers is determined and discussed. The influence of the rotation rate on the generation and stability of secondary branches is demonstrated. Multistability is typical in the parameter range considered.}, language = {en} }